Warmth and connectivity of neighborhood complexes of graphs

In this paper we study a pair of numerical parameters associated to a graph $G$. One the one hand, one can construct $\text{Hom}(K_2, G)$, a space of homomorphisms from a edge $K_2$ into $G$ and study its (topological) connectivity. This approach dates back to the neighborhood complexes introduced by Lov\'asz in his proof of the Kneser conjecture. In another direction Brightwell and Winkler introduced a graph parameter called the warmth $\zeta(G)$ of a graph $G$, based on asymptotic behavior of $d$-branching walks in $G$ and inspired by constructions in statistical physics. Both the warmth of $G$ and the connectivity of $\text{Hom}(K_2,G)$ provide lower bounds on the chromatic number of $G$. Here we seek to relate these two constructions, and in particular we provide evidence for the conjecture that the warmth of a graph $G$ is always less than three plus the connectivity of $\text{Hom}(K_2, G)$. We succeed in establishing a first nontrivial case of the conjecture, by showing that $\zeta(G) \leq 3$ if $\text{Hom}(K_2,G)$ has an infinite first homology group. We also calculate warmth for a family of `twisted toroidal' graphs that are important extremal examples in the context of $\text{Hom}$ complexes. Finally we show that $\zeta(G) \leq n-1$ if a graph $G$ does not have the complete bipartite graph $K_{a,b}$ for $a+b=n$. This provides an analogue for a similar result in the context of $\text{Hom}$ complexes.